Let v be a vector space and let w be a. nonempty subset of v. then w is a subspace of v if and only if the following conditions hold (a) if u and v are in w, then uv is in w. (b) if u is in w and c is a scalar, then cu is in w. use the theorem above to determine whether w is a subspace of v. v f, w (fin f : f(-x) = -f(x)} w is a subspace of v w is not a subspace of v.